Separate the bok choy into stalks. Wash bok choy in several changes of cold water and allow to thoroughly drain in a colander. Trim 1/4-inch from the bottom of each stalk. Halve each stalk lengthwise and cut bok choy into 2-inch-long pieces. In a bowl combine the broth, oyster sauce, soy sauce, tapioca starch, and sugar.
Heat a wok over high heat until hot but not smoking. Add 11/2 teaspoons sesame oil and ginger, and stir-fry for 10 seconds, or until ginger is fragrant. Add remaining 11/2 teaspoons sesame oil, bok choy, and garlic, and stir-fry 1 to 2 minutes, or until leaves are just limp and bok choy is bright green. Stir the broth mixture and swirl into wok.
Stir-fry for 1 to 2 minutes, or until the sauce has thickened slightly and lightly coats the vegetables. Serve immediately.